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4
3
2
1
Grade 12 - Exp.
/ Annales graphes
9 exercises corrected
a
Graph
(2 exercices)
Probabilistic graph
(6 exercices)
Weighted and probabilistic graph
(1 exercice)
1.
Gr
aph
E.7672
A
gardener
has
to
decorate
a
pri-
v
ate
garden
b
y
distributing
10
v
arieties
of
flo
w
ers
noted
V
1
to
V
10
in
differen
t
b
eds.
Some
of
these
v
arieties
cannot
b
e
plan
ted
together
for
v
arious
reasons
(sizes,
c
olors,
climatic
c
onditions,.
.
.
)
and
these
incompatibilities
are
summarized
in
the
table
b
elo
w
(a
cr
oss
indic
ates
inc
omp
atibility
b
etwe
en
two
varieties)
.
Fleur
V
1
V
2
V
3
V
4
V
5
V
6
V
7
V
8
V
9
V
10
V
1
×
×
×
V
2
×
×
×
×
V
3
×
×
×
×
V
4
×
×
×
×
×
V
5
×
×
×
×
V
6
×
×
×
V
7
×
×
V
8
×
×
×
V
9
×
×
V
10
×
×
1
Represen
t
b
y
its
graph
G
the
situation.
2
a
Find
a
complete
subgraph
of
order
4
and
dra
w
it.
b
What
can
w
e
deduce
ab
out
the
coloring
of
the
graph
G
?
What
is
the
minim
um
n
um
b
er
of
b
eds
the
gardener
m
ust
decorate?
3
a
Arrange
the
v
ertices
of
G
in
descending
order
of
de-
gree.
b
Deduce
a
frame
for
C
,
c
hromatic
n
um
b
er
of
G
.
4
a
Pro
ceed
to
the
graph
colariation
G
.
b
What
can
w
e
deduce
for
the
n
um
b
er
C
?
Justify
care-
fully
.
c
Prop
ose
a
set
of
flo
w
erb
eds
with
a
suitable
distribution
of
flo
w
er
v
arieties.
E.7674
A
gardener
o
wns
a
sunn
y
plot
of
land
with
a
shadier
section.
He
decides
to
organize
plots
there
o
where
he
will
plan
t
8
v
arieties
of
v
egetables
:
garlic
(A)
,
squash
(Co)
,
cabbage
(Ch)
,
leeks
(Px)
,
p
eas
(Po)
,
p
otato
es
(Pt)
,
radishes
(R)
and
tomato
es
(T)
.
He
consults
an
almanac
où
listing
plan
t
incompatibilities,
giv
en
b
y
the
t
w
o
tables
:
Exp
ositions
incompatibles
de
plan
tes
Plan
ts
in
par-
tial
shade
Plan
ts
in
full
sun
w
eigh
t
Radishes
Cabbage
T
omato
es
Squash
F
or
example
:
p
eas
are
in-
compatible
with
cabbage,
tomato
es
and
squash.
Asso
ciations
of
incompati-
ble
plan
ts
in
the
same
par-
celle
w
eigh
t
garlic,
leeks
p
otato
es
squash,
radishes
and
tomato
es
cabbage
tomato
es,
garlic,
leeks
and
squash
squash
tomato
es
F
or
example
:
p
eas
are
incompatible
with
garlic
and
p
oireaux
T
o
accoun
t
for
these
incompatibilities
the
gardener
decides
to
mo
del
the
situation
as
an
eigh
t-v
ertex
graph,
with
eac
h
v
ertex
represen
ting
a
v
egetable.
1
On
the
attac
hed
sheet
:
complete
the
graph
highligh
ting
the
exp
osure
incompatibilities
or
incompatible
asso
cia-
tions
indicated
in
the
t
w
o
tables
ab
o
v
e.
2
Calculate
the
sum
of
the
degrees
of
the
v
ertices
of
the
graph,
and
deduce
the
n
um
b
er
of
its
edges
3
Find
a
complete
subgraph
of
order
4
,
what
do
w
e
deduce
for
the
c
hromatic
n
um
b
er
of
the
graph?
4
Giv
e
the
c
hromatic
n
um
b
er
of
the
graph
and
in
terpret
it
as
the
minim
um
n
um
b
er
of
plots
the
gardener
will
ha
v
e
to
create.
5
Giv
e
a
distribution
of
plan
ts
p
er
plot
so
that
eac
h
plot
con
tains
exactly
t
w
o
t
yp
es
of
plan
t
and
the
n
um
b
er
of
plots
is
a
minim
um.
6
Giv
e
a
distribution
of
plan
ts
so
that
a
plot
con
tains
three
plan
ts
and
the
n
um
b
er
of
plots
is
minim
um.
2.
Pr
ob
abilistic
gr
aph
E.6399
The
sales
departmen
ts
of
a
fo
o
d
sup
erstore
ha
v
e
defined
a
customer
profile
that
has
b
een
called
ˇ
consommateur
bio
ı.
Based
on
observ
ations
made
in
previous
y
ears,
it
w
as
found
that
:
90
%
of
customers
ˇ
consumer
bio
ı
main
tained
this
prac-
tice
the
follo
wing
y
ear
;
15
%
of
customers
who
do
not
ha
v
e
the
profile
of
ˇ
con-
sumer
bio
ı
en
tered
the
ˇ
consumer
bio
ı
category
the
fol-
lo
wing
y
ear.
This
ev
olution
is
assumed
to
con
tin
ue
y
ear
on
y
ear
from
2013
,
the
y
ear
in
whic
h
20
%
customers
w
ere
found
to
ha
v
e
the
pro-
file
ˇ
consommateur
bio
ı.
By
a
random
dra
w
ev
ery
y
ear,
w
e
c
ho
ose
a
customer
of
this
departmen
t
store.
F
or
an
y
natural
n
um
b
er
n
,
note
:
b
n
:
the
probabilit
y
that
the
customer
c
hosen
in
the
y
ear
2013+
n
is
a
ˇ
consumer
bio
ı
;
c
n
:
the
probabilit
y
that
the
customer
c
hosen
in
the
y
ear
2013+
n
is
not
a
ˇ
consumer
bio
ı.
P
n
:
the
ro
w
matrix
b
n
c
n
giving
the
probabilistic
state
in
the
y
ear
2013+
n
.
1
a
Represen
t
the
situation
b
y
a
probabilistic
graph
with
v
ertices
B
and
C
où
B
corresp
onds
to
the
state
ˇ
con-
h
ttps:/
/c
hingmath.fr
sacados/7672
Antilles-Guyane Juin 2006
sacados/7674
chapExoCorrec/6399
sacados/6399
Antilles-Guyane Juin 2014
sumer
bio
ı.
b
Giv
e
P
0
the
probabilistic
state
in
2013
and
the
transi-
tion
matrix
M
corresp
onding
to
this
graph,
with
the
v
ertices
B
and
C
rank
ed
in
that
order.
c
W
e
giv
e
the
matrix
M
2
:
M
2
=
0.825
0.175
0.2625
0.7375
Sp
ecifying
the
calculation
metho
d,
determine
the
prob-
abilit
y
that
the
customer
c
hosen
in
2015
is
a
ˇ
bio
ı
con-
sumer.
d
Determine
the
stable
state
b
c
of
the
probabilistic
graph.
2
The
sup
ermark
et
manager
claims
that,
in
the
near
fu-
ture,
more
than
half
his
customers
will
ha
v
e
the
profile
ˇ
organic
consumer
ı.
a
Cop
y
and
complete
the
follo
wing
algorithm
so
that
at
the
end
of
its
execution
the
v
ariable
N
has
the
v
alue
of
the
minim
um
n
um
b
er
of
y
ears
for
the
director’s
state-
men
t
to
b
e
v
erified
:
N
←
0
B
←
0.2
C
←
0.8
As
long
as
...
B
←
0.9
×
B
+
0.15
×
C
C
←
1
−
B
N
←
N+1
End
As
long
as
E.6401
Alice
tak
es
part
in
an
arc
hery
comp
etition
;
she
mak
es
sev
eral
arro
w
thro
ws.
When
she
hits
the
target
on
one
thro
w,
the
probabilit
y
that
she
will
hit
the
target
on
the
next
thro
w
is
equal
to
0.9
.
When
she
has
missed
the
target
on
one
thro
w,
Alice
b
ecomes
unfo
cused
and
the
probabilit
y
that
she
will
hit
the
target
on
the
next
thro
w
is
equal
to
0.4
.
It
is
assumed
that
on
the
first
thro
w
she
has
as
m
uc
h
c
hance
of
hitting
the
target
as
of
missing
it.
F
or
an
y
strictly
p
ositiv
e
natural
n
um
b
er
n
,
note
:
a
n
the
probabilit
y
of
Alice
hitting
the
target
on
the
n
-th
thro
w
;
b
n
the
probabilit
y
that
Alice
misses
the
target
on
the
n
-th
thro
w
;
P
n
=
a
n
b
n
the
ro
w
matrix
translating
the
proba-
bilistic
state
at
the
n
-th
thro
w.
1
a
Represen
t
the
situation
b
y
a
probabilistic
graph
with
v
ertices
A
and
B
(
A
r
epr
esenting
the
state
ˇ
Alice
reac
hes
the
cible
ı
and
B
state
ˇ
Alice
misses
her
cible
ı)
.
b
State
the
transition
matrix
M
asso
ciated
with
this
graph.
The
v
ertices
A
and
B
will
b
e
tak
en
in
order
(
A
;
B
)
.
c
Justify
that
:
P
1
=
0.5
0.5
;
P
2
=
0.65
0.35
2
a
Sho
w
that,
for
an
y
strictly
p
ositiv
e
in
teger
n
:
a
n
+1
=
0.9
·
a
n
+
0.4
·
b
n
b
Deduce
that,
for
an
y
strictly
p
ositiv
e
in
teger
n
:
a
n
+1
=
0.5
·
a
n
+
0.4
3
a
Consider
b
elo
w
the
function
f
of
an
algorithm
tak-
ing
as
parameter
the
argumen
t
n
whic
h
is
a
natural
n
um
b
er
greater
than
or
equal
to
2
.
Function
f(n)
a
←
0.5
b
←
0.5
For
i
ranging
from
2
to
n
a
←
...
×
a+
...
b
←
1
−
a
End
To
Renvoyer
(
a
;
b
)
Complete
the
function
f
so
that
at
the
end
of
its
exe-
cution,
the
pair
returned
b
y
this
function
indicates
the
probabilistic
state
at
the
n
-th
thro
w.
b
Determine
the
pair
of
v
alues
returned
when
this
func-
tion
is
called
with
the
argumen
t
n
=
5
.
4
a
Consider
the
sequence
u
n
defined
for
an
y
strcite-
men
t
p
ositiv
e
natural
n
um
b
er
n
b
y
:
u
n
=
a
n
−
0.8
.
Sho
w
that
the
sequence
u
n
is
a
geometric
sequence
whose
reason
and
first
term
will
b
e
sp
ecified.
b
Giv
e
the
expression
of
u
n
as
a
function
of
n
,
then
de-
duce
that
for
an
y
strictly
p
ositiv
e
natural
n
um
b
er
n
:
a
n
=
0.8
−
0.3
×
0.5
n
−
1
.
c
In
the
long
term,
what
can
w
e
think
ab
out
the
proba-
bilit
y
of
Alice
hitting
the
target?
d
By
what
other
metho
d
w
ould
w
e
ha
v
e
found
the
pre-
vious
result?
h
ttps:/
/c
hingmath.fr
chapExoCorrec/6401
sacados/6401
E.6403
T
o
satisfy
its
mem
b
ers,
a
sp
orts
club
has
in
tro
duced
three
learning
lev
els
:
DEBUT
ANT
(D),
CONFIRMED
(C)
and
EXPER
T
(E)
As
of
1
er
Septem
b
er
2012
,
at
registration,
the
club
had
:
30
%
of
b
eginners
;
50
%
of
confirmed
;
20
%
of
exp
erts.
F
rom
one
y
ear
to
the
next,
w
e
note
that
:
among
b
eginner-lev
el
mem
b
ers,
40
%
remain
at
this
lev
el
and
60
%
mo
v
e
up
to
the
confirmed
lev
el
;
among
confirmed
mem
b
ers,
60
%
remain
at
this
lev
el
and
40
%
mo
v
e
up
to
exp
ert
lev
el
;
among
exp
ert
lev
el
mem
b
ers,
80
%
remain
at
this
lev
el,
10
%
mo
v
e
do
wn
to
the
confirmed
lev
el,
and
the
remain-
ing
10
%
prefer
to
return
to
the
basics
at
the
b
eginner
lev
el.
W
e
consider
that
there
are
no
new
comers
or
departures
in
the
club.
Let
P
n
=
d
n
c
n
e
n
b
e
the
ro
w
matrix
describing
the
probabilistic
state
of
the
distribution
among
the
three
learn-
ing
lev
els
D
,
C
and
E
at
1
Septem
b
er
1
in
the
y
ear
2012+
n
for
an
y
natural
n
um
b
er
n
.
1
a
Giv
e
without
justification
the
matrix
P
0
.
b
T
ranslate
the
situation
b
y
a
probabilistic
graph
of
v
er-
tices
D
,
C
and
E
.
W
e
giv
e
the
square
matrix
M
of
transition
resp
ecting
the
or-
der
D
,
C
,
E
of
the
v
ertices.
M
=
0.4
0
.
6
0
0
0.6
0.4
0.1
0.1
0
.
8
In
the
rest
of
the
exercise,
w
e
can
use
the
follo
wing
results
(r
esults
r
ounde
d
to
the
thousandth)
:
M
5
=
0.085
0.331
0.584
0.097
0.293
0.610
0.104
0.298
0.598
M
10
=
0.100
0.299
0.601
0.100
0.300
0.600
0.100
0.300
0.600
2
In
this
matrix,
w
e
read
0
.
6
and
0
.
8
in
b
old
italics.
a
Sp
ecify
,
using
a
sen
tence,
what
these
t
w
o
v
alues
corre-
sp
ond
to
in
relation
to
the
situation
studied.
b
Calculate
P
1
.
c
Determine
the
predicted
distribution,
in
p
ercen
tages,
of
mem
b
ers
in
this
sp
orts
club
at
1
er
Septem
b
er
2017
.
Results
will
b
e
giv
en
to
the
nearest
0.1
%
.
3
a
By
calculating
P
10
,
mak
e
a
conjecture
ab
out
the
ma-
trix
P
corresp
onding
to
the
stable
probabilistic
state.
b
V
erify
this
conjecture.
c
What
conclusion
can
b
e
dra
wn
for
the
distribution
of
mem
b
ers?
E.6404
In
the
first
w
eek
of
the
y
ear,
the
comm
unications
manager
of
a
large
compan
y
asks
emplo
y
ees
to
decide
on
a
new
logo,
with
the
c
hoice
to
b
e
made
b
y
a
v
ote
at
the
end
of
the
y
ear.
T
w
o
logos,
designated
A
and
B
resp
ectiv
ely
,
are
put
to
the
v
ote.
A
t
the
presen
tation,
whic
h
tak
es
place
in
the
first
w
eek
of
the
y
ear,
24
%
of
emplo
y
ees
fa
v
or
the
A
logo
and
all
other
emplo
y
ees
fa
v
or
the
B
logo.
Discussions
b
et
w
een
emplo
y
ees
ev
olv
e
this
breakdo
wn
throughout
the
y
ear.
Th
us,
9
%
of
emplo
y
ees
in
fa
v
or
of
the
A
logo
c
hange
their
minds
the
follo
wing
w
eek,
and
16
%
of
emplo
y
ees
in
fa
v
or
of
the
B
logo
c
hange
their
minds
the
follo
wing
w
eek.
F
or
an
y
n
,
n
1
,
w
e
note
:
a
n
the
probabilit
y
of
an
emplo
y
ee
fa
v
oring
the
A
logo
in
the
w
eek
n
;
b
n
the
probabilit
y
of
an
emplo
y
ee
fa
v
oring
the
B
logo
in
w
eek
n
;
P
n
the
matrix
a
n
b
n
translating
the
probabilistic
state
the
w
eek
n
.
W
e
therefore
ha
v
e,
for
an
y
n
1
:
a
n
+
b
n
=
1
;
P
1
=
0.24
0.76
1
T
ranslate
the
situation
in
to
a
probabilistic
graph
with
v
ertices
A
and
B
.
2
Determine
the
transition
matrix
M
of
this
graph,
arrang-
ing
the
v
ertices
in
alphab
etical
order.
3
a
Using
the
relationship
P
n
+1
=
P
n
×
M
,
express,
for
an
y
n
1
,
a
n
+1
as
a
function
of
a
n
and
b
n
.
b
Deduce
that
for
an
y
n
1
:
a
n
+1
=
0.75
·
a
n
+
0.16
.
4
Using
the
calculator,
giv
e,
without
justification,
the
prob-
abilit
y
to
within
0.001
that
an
emplo
y
ee
will
b
e
in
fa
v
or
of
the
A
logo
the
w
eek
4
.
5
W
e
note
P
=
a
b
the
stable
state
of
the
emplo
y
ee
dis-
tribution.
a
Determine
a
system
of
t
w
o
equations
that
a
and
b
m
ust
v
erify
.
b
Solv
e
the
system
obtained
in
the
previous
question.
c
The
stable
state
is
assumed
to
b
e
P
=
0.64
0.36
.
In
terpret
the
result.
6
Consider
the
follo
wing
algorithm
:
A
←
0.24
N
←
0
As
long
as
A
<
0.639
N
←
N+1
A
←
0.75
×
A+0.16
End
As
long
as
After
running
the
algorithm,
giv
e,
an
in
terpretation
of
the
v
alue
of
the
v
ariable
N
(you
ar
e
not
aske
d
to
give
the
value
of
N
at
the
end
of
exe
cution
of
the
algorithm)
.
h
ttps:/
/c
hingmath.fr
chapExoCorrec/6403
sacados/6403
chapExoCorrec/6404
sacados/6404
E.6408
The
w
ebsites
A
,
B
,
C
ha
v
e
links
b
et
w
een
them.
A
surfer
connected
to
one
of
these
three
sites
can,
ev
ery
few
min
utes,
either
sta
y
there
or
use
a
link
to
one
of
the
other
t
w
o
sites
:
F
or
a
surfer
connected
to
A
,
the
probabilit
y
of
using
the
link
to
B
is
0.2
and
that
of
using
the
link
to
C
is
0.2
.
F
or
an
In
ternet
user
connected
to
B
,
the
probabilit
y
of
using
the
link
to
A
is
0.1
and
that
of
using
the
link
to
C
is
0.4
.
F
or
an
In
ternet
user
connected
to
C
,
the
probabilit
y
of
using
the
link
to
A
is
0.2
but
there
is
no
direct
link
to
B
.
The
unit
of
time
is
the
min
ute,
and
at
an
instan
t
t
=
0
,
the
n
um
b
er
of
visitors
is,
resp
ectiv
ely
on
A
,
B
and
C
:
100
,
0
and
0
.
W
e
represen
t
the
distribution
of
In
ternet
users
on
the
three
sites
after
t
min
utes
b
y
a
matrix
N
t
;
th
us,
N
0
=
100
0
0
.
It
is
assumed
that
there
are
no
disconnections
during
the
hour
of
(fr
om
t
=
0
to
t
=
60
)
or
new
In
ternet
visitors.
1
Represen
t
the
probabilistic
graph
of
v
ertices
A
,
B
and
C
corresp
onding
to
the
situation
describ
ed.
2
W
rite
the
transition
matrix
M
asso
ciated
with
this
graph
(in
or
der
A
,
B
,
C
)
.
3
W
e
giv
e
:
M
2
=
0.42
0.22
0.36
0.19
0.27
0.54
0.28
0.04
0.68
M
20
≈
0.3125
0.125
0.5625
0.3125
0.125
0.5625
0.3125
0.125
0.5625
Calculate
N
2
.
In
terpret
the
result
obtained.
4
Calculate
N
0
×
M
20
.
Conjecture
the
steady-state
v
alue
and
in
terpret
the
answ
er.
5
One
of
the
In
ternet
users
transmits
a
virus
to
an
y
site
he
visits.
He
initially
connects
to
C
and
starts
bro
wsing.
A
t
time
t
=
0
,
site
C
is
therefore
infected.
a
What
is
the
probabilit
y
that
at
time
t
=
1
site
A
is
infected?
b
What
is
the
probabilit
y
that
at
time
t
=
2
all
three
sites
are
infected?
E.6421
In
one
coun
try
,
only
t
w
o
mobile
op
erators
SAFIR
and
TECIM
offer
4G
(data
tr
ansmission
standar
d)
.
One
study
sho
w
ed
that
from
one
y
ear
to
the
next
:
41
%
of
customers
of
op
erator
S
AF
I
R
lea
v
e
it
for
op
era-
tor
TECIM
;
9
%
of
the
TECIM
op
erator’s
customers
lea
v
e
it
for
the
SAFIR
op
erator
;
No
customers
giv
e
up
using
4G.
This
situation
can
b
e
mo
deled
b
y
a
probabilistic
graph
G
with
v
ertices
S
and
T
où
:
S
is
the
ev
en
t
ˇ
the
4G
user
is
a
customer
of
the
op
erator
SAFIR
ı
;
T
is
the
ev
en
t
ˇ
the
4G
user
is
a
customer
of
the
op
erator
TECIM
ı
;
W
e
note
P
n
=
s
n
t
n
the
ro
w
matrix
of
the
probabilistic
state
for
the
y
ear
2014+
n
.
In
this
exercise,
w
e
prop
ose
to
find
out
whether
the
op
erator
TECIM
will
ac
hiev
e
the
ob
jectiv
e
of
ha
ving
as
customers
at
least
80
%
of
the
p
opulation
using
4G.
P
art
A
1
Dra
w
the
probabilistic
graph
G
.
2
W
e
admit
that
the
transition
matrix
of
the
graph
G
considering
the
v
ertices
in
the
order
S
and
T
is
M
=
0.59
0.41
0.09
0.91
W
e
denote
P
=
a
b
the
ro
w
matrix
corresp
onding
to
the
stable
state
of
this
graph
G
.
a
Sho
w
that
the
n
um
b
ers
a
and
b
are
solutions
of
the
system
:
0.41
·
a
−
0.09
·
b
=
0
a
+
b
=
1
b
Solv
e
the
previous
system.
3
Assume
a
=
0.18
and
b
=
0.82
.
Determine,
with
justification,
whether
the
TECIM
op
er-
ator
can
exp
ect
to
ac
hiev
e
its
ob
jectiv
e.
P
art
B
In
2014
,
w
e
kno
w
that
35
%
of
4G
users
are
customers
of
op-
erator
SAFIR
and
65
%
are
customers
of
op
erator
TECIM.
Th
us
:
P
0
=
0.35
0.65
.
1
Determine
the
distribution
of
4G
customers
after
2
y
ears.
2
Sho
w
that,
for
an
y
natural
n
um
b
er
n
,
w
e
ha
v
e
:
t
n
+1
=
0.5
·
t
n
+
0.41
3
T
o
find
out
ho
w
man
y
y
ears
it
will
tak
e
the
TECIM
op-
erator
to
reac
h
his
goal,
w
e
first
dev
elop
ed
the
algorithm
b
elo
w.
Cop
y
and
complete
the
lines
‘
.4
and
‘
.5
so
that
at
the
end
of
execution
the
v
alue
of
the
v
ariable
N
giv
es
the
exp
ected
result.
‘
.1
T
←
0.65
‘
.2
N
←
0
‘
.3
As
long
as
T
<
0.80
‘
.4
T
←
...
‘
.5
N
←
...
‘
.6
End
As
long
as
4
Consider
the
sequence
u
n
defined
for
an
y
natural
n
um-
b
er
n
b
y
:
u
n
=
t
n
−
0.82
.
a
Sho
w
that
the
sequence
u
n
is
a
geometric
sequence
of
reason
0.5
.
Sp
ecify
its
first
term.
b
Deduce
that
:
t
n
=
−
0.17
×
0.5
n
+
0.82
c
Solv
e
in
the
set
of
natural
n
um
b
ers
the
inequation
:
−
0.17
×
0.5
n
+
0.82
0.80
.
d
In
terpret
this
result
in
the
con
text
of
the
statemen
t.
h
ttps:/
/c
hingmath.fr
chapExoCorrec/6408
sacados/6408
chapExoCorrec/6421
sacados/6421
Liban Mai 2015
3.
W
eighte
d
and
pr
ob
abilistic
gr
aph
E.6411
A
study
is
carried
out
eac
h
win
ter
on
a
p
opulation
made
up
of
p
eople
who
can
practice
do
wnhill
skiing
or
sno
wb
oarding.
The
studen
t
rev
eals
that
:
If
a
p
erson
practices
do
wnhill
skiing,
then
the
probabilit
y
that
he
or
she
will
practice
sno
wb
oarding
the
follo
wing
win
ter
is
equal
to
0.2
.
If
a
p
erson
practices
sno
wb
oarding,
then
the
probabilit
y
that
he
or
she
will
practice
do
wnhill
skiing
the
follo
wing
win
ter
is
equal
to
0.3
.
W
e
note
S
the
state
:
ˇ
the
p
erson
go
es
do
wnhill
skiing
ı
and
S
the
state
:
ˇ
the
p
erson
practices
sno
wb
oard
ı.
W
e
also
note
for
an
y
natural
n
um
b
er
n
:
p
n
the
probabilit
y
that
a
p
erson
will
practice
do
wnhill
skiing
in
the
n
-th
win
ter
;
q
n
the
probabilit
y
of
a
p
erson
sno
wb
oarding
in
the
n
-th
win
ter
;
P
n
=
p
n
q
n
the
ro
w
matrix
giving
the
probabilistic
state
of
the
system
during
the
n
-th
win
ter.
The
initial
p
opulation
is
assumed
to
comprise
only
do
wnhill
skiers,
so
w
e
ha
v
e
P
0
=
1
0
.
P
art
A
1
Represen
t
the
situation
using
a
probabilistic
graph
of
v
er-
tices
S
and
S
.
2
a
Giv
e
the
transition
matrix
M
of
this
probabilistic
graph.
b
Calculate
M
2
.
c
Determine
probabilistic
state
P
2
.
3
Sho
w
that
for
an
y
natural
n
um
b
er
n
,
w
e
ha
v
e
:
p
n
+1
=
0.5
·
p
n
+
0.3
.
4
Consider
the
follo
wing
algorithm
:
V
ariables
‘
.
1
J
and
N
are
natural
n
um
b
ers
‘
.
2
p
is
a
real
n
um
b
er
En
trée
‘
.
3
Seize
N
Initialization
‘
.
4
p
tak
es
the
v
alue
1
T
raitemen
t
‘
.
5
F
or
J
ranging
from
1
to
N
‘
.
6
p
tak
es
the
v
alue
.
.
.
.
.
.
.
.
.
‘
.
7
Fin
P
our
Sortie
‘
.
8
Affic
her
p
Cop
y
and
complete
the
line
6
of
this
algorithm
to
obtain
the
probabilit
y
p
n
.
P
art
B
Consider,
for
an
y
natural
n
um
b
er
n
,
the
ev
en
t
S
n
:
ˇ
the
p
er-
son
practices
do
wnhill
skiing
during
the
n
-th
hiv
er
ı.
The
probabilit
y
of
the
ev
en
t
S
n
is
denoted
p
S
n
.
W
e
there-
fore
ha
v
e
:
p
n
=
p
S
n
.
W
e
kno
w
from
part
A
that
for
an
y
natural
n
um
b
er
n
:
p
n
+1
=
0.5
·
p
n
+
0.3
Let
u
n
b
e
the
sequence
defined
for
an
y
natural
n
um
b
er
n
b
y
:
u
n
=
p
n
−
0.6
.
1
Sho
w
that
the
sequence
u
n
is
a
geometric
sequence
of
reason
0.5
and
sp
ecify
the
v
alue
of
u
0
.
2
Deduce
the
expression
of
u
n
as
a
function
of
n
and
then
the
expression
of
p
n
as
a
function
of
n
.
3
Determine
the
limit
of
the
sequence
p
n
and
in
terpret
the
result.
P
art
C
P
art
of
the
ski
area
is
represen
ted
b
y
the
graph
b
elo
w.
V
ertex
A
represen
ts
the
top
of
the
ski
runs
and
v
ertex
I
rep-
resen
ts
the
b
ottom.
V
ertices
B
,
C
,
D
,
E
F
,
G
,
and
H
represen
t
crossing
p
oin
ts.
Eac
h
of
the
edges
is
w
eigh
ted
b
y
the
distance,
in
h
undred
meters,
b
et
w
een
t
w
o
v
ertices.
Using
Dijkstra’s
algorithm,
determine
the
minim
um
distance
to
connect
v
ertex
A
to
v
ertex
I
.
h
ttps:/
/c
hingmath.fr
chapExoCorrec/6411
sacados/6411
A
B
C
D
E
F
G
H
I
7
16
21
13
18
15
8
12
8
5
6
5
12
18
7
13
7
19
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